# Electric Potential Energy

by Jared Rovny

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00:01 As a summary of the electric potential and how it's related to voltage, we're going to see more of these ideas pop up as we go, but let's go to the electric potential energy.

00:09 The electric potential energy is related to the electric potential field because what we do to get the electric potential energy is to multiply by whatever charge we have placed in the field.

00:23 In other words, we saw that for the scalar electric potential field, the units were joules per coulomb or how much energy was there per unit charge that I introduced into the field.

00:34 What we do in this case is simply to add to that charge this other q and so you can see in the equation for the electric potential energy we now have two factors of q, one from the charge in the center that is causing this electric potential field and the other one for the charge that's introduced and put into this electric potential field.

00:53 We represent the electric potential energy also by the letter U, as we did with the other potential energies from gravity as well as from springs.

01:02 Again, this quantity is going to be a scalar just like the electric potential and not a vector.

01:09 We also have to be careful with the names here.

01:11 This field is called the electric potential energy which is different from the field that we just introduced called the phi which was just the electric potential where we measured voltages as differences in the electric potential from one point to another.

01:27 The units of the electric potential energy are in fact just units of energy.

01:33 It is joules because what we're measuring is how much energy it would take to introduce a new charge close to this already existing charge.

01:40 This is exactly analogous with the earth.

01:44 So, this might be a helpful analogy for you to keep in your head.

01:47 The earth, we already introduced the gravitational potential energy, so that if we had a mass somewhere near the earth we could say that that mass or the mass with the earth's system has some sort of a gravitational potential energy.

02:01 The way we think about this gravitational potential energy is very similar to the way the potential energy near the earth behaves.

02:10 So, we have an equation for the electric potential energy which is listed up at the top here but we also had an equation for the gravitational potential energy.

02:19 So, when we lift a mass some distance above the earth, we say that it has some potential energy to it or that the earth mass system has some potential energy to it.

02:30 This will behave the same way in a sense that the mass is going to want to fall closer to the earth or two electric charges are going to want to minimize the energy between them and go closer to each other.

02:41 There are some differences so you don't want to take this analogy too far but it's helpful in a way that you think about it.

02:47 For, example when we were talking about things near the surface of the earth, we were often defining an arbitrary zero point for a given coordinate system so we might define those differently but we we're also saying that electric forces can attract or repel which is very different from gravity which is always only attracting two objects rather than ever repelling them.

03:06 The electric potential energy is something we could also plot on a graph.

03:12 So for example, on the bottom right here what I left rather, what I have, is a graph whose vertical access is the electric potential energy and then whose horizontal axis is just exactly the distance between the two objects.

03:24 If I plot the electric potential energy difference for two oppositely charged objects, you can look at the equation for the electric potential energy and see that the numerator which has these two charges together, one positive and one negative, will always have a negative sign and so you can see this red curve where we've plotted the electric potential energy is always going to be negative or below the axis.

03:47 The value of this negative electric potential energy is going to get closer and closer to zero as we get further and further away.

03:55 In other words, if you looked at the radius term in this electric potential energy, if we increased that r, if we increased the distance, the overall number for the electric potential energy will get smaller and smaller and smaller.

04:08 If these two charges were instead the same sign chart maybe both positive charges and we asked ourselves about the electric potential energy, it would follow the exact same kind of shape except not be negative, it would be positive and so we could flip it to the top.

04:22 The way objects move in an electric potential energy field is very important to understand in terms of these graphs.

04:30 So, in these graphs we can see that charges are always going to try to move towards more negative or less electric potential energy.

04:38 So for example, the positive charge that's placed in the presence of the negative charge wants to sort of fall down this red slope that we've drawn, it wants to go to our lower and lower electric potential energy whereas for two positive charges that are going to repel each other, the positive charge, maybe the second one was initially close to the first one, is going to sort of go downhill or towards more less and less electric potential energy.

05:02 This is a very important concept to know because if sometimes you're presented with simply a graph of the electric potential energy, you should always know that an object is trying to minimize and go to a lower electric potential energy.

05:17 We can also describe our electric potential energy as we briefly discussed, in terms of the electric potential field and again keep in your mind distinct the idea of the electric potential field from the electric potential energy.

05:31 What we said with the electric field was that it was measuring the joules per coulomb or how much energy did we need per amount of charge that we were going to introduce near another charge.

05:43 We also saw from our equations that the electric potential energy was simply equal to that charge, whichever one we introduced, times the potential.

05:52 So, if we have a difference in potential energy, so for example, we have two locations near this negative charge here.

06:00 We have a difference in potential energy which will be related then to the charge times the difference in the electric potential.

06:08 Therefore the change in potential energy and I know this can get confusing so be very careful here.

06:13 The change in potential energy is equal to q, the charge, times the change in the potential but we already said what the change in potential is.

06:22 The change in potential is what we call the voltage and so therefore the change in the potential energy of a charge moving from one point to another will be equal to the magnitude of that charge times the voltage difference between those two points.

06:36 Taking this a little further we can simply say it in as many words that the difference in energy between two points near a charge like this negative one would be that charge times the change in voltage of where that charge moved and finally, we can say that the energy is gained or lost due to force experienced by that charge or to put it in another way we said that an object can do work by applying a force over a distance.

07:02 So therefore when an object like this q, this positive charge, changes its position it might gain or lose energy just like an object I was lifting away from the earth or towards the earth can gain or lose gravitational potential energy and this energy is gained or lost because this negative charge is applying a force over a distance and therefore doing work.

07:23 This electric field is also what we would call a conservative field.

07:28 If you remember our discussion of gravity and the definition of a conservative force we could say the same thing about the electric force.

07:36 It's conservative in the sense that it only converts if you will, the energy between kinetic and potential.

07:43 It's never taking away energy or giving energy, it's only converting between the two different types of energy.

07:49 This means that the energy difference between two points will only depend on the final and conserve, sorry, the final and initial position of your charge and so for that reason, the electric force is also a conservative force.

08:04 Following from this definition that the electric force is also a conservative force, we can measure the work done by the electric field.

08:11 Because it's a conservative force, the work done will only be related to the difference of the potential energy between the initial and final locations of your charge rather than the path that the charge took getting from one point to the other.

08:25 What I have here is a table summarizing all of these different fields that we've just introduced.

08:32 It can be kind of complicated to keep everything straight because we have now introduced four different kinds of fields.

08:38 We have the electric force, the electric field, the electric potential energy, and the electric potential.

08:46 What you can do is think of these all sort of together.

08:48 What I've done is in this table, separated things vertically you can see we have vectors with values of 1 over r squared in the term and scalars with values of 1 over r in the term.

08:59 Also, by coulombs we can say that some of these terms including the force and the potential energy can only be measured for two charges.

09:07 For example, you can measure the force between two charges or the amount of potential energy stored by two charges being near each other whereas for other, the other two equations on the right here.

09:17 We have expressions for a single charge.

09:20 For example, the field and the potential can be defined as the field or potential from a single charge placed in the center of your coordinate system and so thinking of all these together, you can see some similarities and some differences.

09:33 The important thing is that you really look at this table, understand the differences in these four terms, and then very importantly are able to convert between these four terms in a physical scenario or in a given problem.

09:46 The most common things you'll have to do in a problem are to convert from the electric field to the amount of force experienced by a charge in that electric field and you can always find this simply by multiplying the electric field by the charge q and so we have this first equation that the force is equal to the charge q times the electric field that that charge is being placed in.

10:07 The second most common equation that you'll be needing to use to convert between these is to known that the change in potential energy that a charge experiences in going from one place to another is equal to the magnitude of that charge times the change in the potential field that it experiences going from one place to another and again we call that change in potential the voltage and so the change in energy that a charge has in going from one place to another is equal to the value of that charge times the voltage difference in the two locations that it's going to inhabit from initial to final.

10:40 With this table we have a summary, a good summary of the fields and forces that we've introduced as well as how to relate between those.

10:47 Now, we're ready to go on to circuits as well as magnetism.

10:51 Thanks for listening.

### About the Lecture

The lecture Electric Potential Energy by Jared Rovny is from the course Electrostatics.

### Included Quiz Questions

1. Joules
2. Newtons
3. Coulombs
4. Joules per Coulomb
5. Newtons per Coulomb
1. Charges move towards more negative energy.
2. Charges move towards more positive energy.
3. Charges move to smaller magnitudes in energy.
4. Charges move to maximize potential energy.
5. Charges move away from other charges with energy.
1. Multiply the voltage by the charge
2. Divide the voltage by the charge
3. Multiply the voltage by the energy
4. Multiply the charge by the Force
5. Multiply the voltage by the electric potential
1. They were the same
2. The longer path
3. The shorter path
4. More information is needed
5. The longer path if the charge was negative
1. Potential energy and electric potential
2. Potential energy and electric force
3. Potential energy and electric field
4. Electric field and electric force
5. Electric Force and voltage

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